Introduction Abacus Counting-rods Conclusion Programming Languages for Pre-Mechanical Calculating Tools Baptiste Mélès 梅乐思 ArchivesHenriPoincaré–UniversitédeLorraine(Nancy南锡,France) “Cultures of Mathematics and Logic”, Guangzhou 广州, China, November 2012 1/51 BaptisteMélès ProgrammingLanguagesforCalculatingTools Introduction “Manisananimalwhothinkswithhisfingers” Abacus Historyofmathematicsandcomputerscience Counting-rods Anexampleofstructuralanthropology: percussiontools Conclusion “Man is an animal who thinks with his fingers” “As for the algebra that Chinese developped by applying it first to the resolution of elementary problems of geometry, it can not be dissociated from an ability to manipulations which is unequally cultivated in the different civilizations. As said Marcel Mauss, quoting Maurice Halbwachs: “Man is an animal who thinks with his fingers” (« L’homme est un animal qui pense avec ses doigts »), a beautiful maxim which warns us against too intellectualist approaches, and which Chinese mathematicians would probably have approved. The history of a civilization implies at once socio-political structures, notions, mental attitudes, behaviours, gestures, and everything that the ethnologists name techniques of the body (« techniques du corps »).” — Jacques Gernet, L’Intelligence de la Chine, “Histoire sociale et intellectuelle”, p. 254. 2/51 BaptisteMélès ProgrammingLanguagesforCalculatingTools Introduction “Manisananimalwhothinkswithhisfingers” Abacus Historyofmathematicsandcomputerscience Counting-rods Anexampleofstructuralanthropology: percussiontools Conclusion Question Initial question: What functions can be computed with different kinds of non-mechanical calculating tools? Examples of calculating tools: abacus, Chinese counting-rods, logarithm tables, slide rule, counting on paper... Examples of limitations: logarithm tables, slide rule: no addition; abacus: no resolution of systems of n linear equations? no logarithm? 3/51 BaptisteMélès ProgrammingLanguagesforCalculatingTools Introduction “Manisananimalwhothinkswithhisfingers” Abacus Historyofmathematicsandcomputerscience Counting-rods Anexampleofstructuralanthropology: percussiontools Conclusion Question Initial question: What functions can be computed with different kinds of non-mechanical calculating tools? Examples of calculating tools: abacus, Chinese counting-rods, logarithm tables, slide rule, counting on paper... Examples of limitations: logarithm tables, slide rule: no addition; abacus: no resolution of systems of n linear equations? no logarithm? 3/51 BaptisteMélès ProgrammingLanguagesforCalculatingTools Introduction “Manisananimalwhothinkswithhisfingers” Abacus Historyofmathematicsandcomputerscience Counting-rods Anexampleofstructuralanthropology: percussiontools Conclusion Question Initial question: What functions can be computed with different kinds of non-mechanical calculating tools? Examples of calculating tools: abacus, Chinese counting-rods, logarithm tables, slide rule, counting on paper... Examples of limitations: logarithm tables, slide rule: no addition; abacus: no resolution of systems of n linear equations? no logarithm? 3/51 BaptisteMélès ProgrammingLanguagesforCalculatingTools Introduction “Manisananimalwhothinkswithhisfingers” Abacus Historyofmathematicsandcomputerscience Counting-rods Anexampleofstructuralanthropology: percussiontools Conclusion Classical answers Two possible answers: A posteriori (history of mathematics): “just read your classics”. Counting-rods: Sunzi 孫子 (ca. 300), Sunzi Suanjing 孫子算 經; Abacus: Cheng Dawei 程大位 (1533–1606), Suanfa Tongzong 算法統宗 (1592). A priori (computer science): “just make a machine”. Alan Turing (1912–1954), “On Computable Numbers” (1936): the definition of “Turing machines” begins with the behaviour of a human “computer” (i.e. the calculating man); Joachim Lambek (born 1922), “How to Program an Infinite Abacus” (1961). 4/51 BaptisteMélès ProgrammingLanguagesforCalculatingTools Introduction “Manisananimalwhothinkswithhisfingers” Abacus Historyofmathematicsandcomputerscience Counting-rods Anexampleofstructuralanthropology: percussiontools Conclusion Classical answers Two possible answers: A posteriori (history of mathematics): “just read your classics”. Counting-rods: Sunzi 孫子 (ca. 300), Sunzi Suanjing 孫子算 經; Abacus: Cheng Dawei 程大位 (1533–1606), Suanfa Tongzong 算法統宗 (1592). A priori (computer science): “just make a machine”. Alan Turing (1912–1954), “On Computable Numbers” (1936): the definition of “Turing machines” begins with the behaviour of a human “computer” (i.e. the calculating man); Joachim Lambek (born 1922), “How to Program an Infinite Abacus” (1961). 4/51 BaptisteMélès ProgrammingLanguagesforCalculatingTools Introduction “Manisananimalwhothinkswithhisfingers” Abacus Historyofmathematicsandcomputerscience Counting-rods Anexampleofstructuralanthropology: percussiontools Conclusion Classical answers Two possible answers: A posteriori (history of mathematics): “just read your classics”. Counting-rods: Sunzi 孫子 (ca. 300), Sunzi Suanjing 孫子算 經; Abacus: Cheng Dawei 程大位 (1533–1606), Suanfa Tongzong 算法統宗 (1592). A priori (computer science): “just make a machine”. Alan Turing (1912–1954), “On Computable Numbers” (1936): the definition of “Turing machines” begins with the behaviour of a human “computer” (i.e. the calculating man); Joachim Lambek (born 1922), “How to Program an Infinite Abacus” (1961). 4/51 BaptisteMélès ProgrammingLanguagesforCalculatingTools Introduction “Manisananimalwhothinkswithhisfingers” Abacus Historyofmathematicsandcomputerscience Counting-rods Anexampleofstructuralanthropology: percussiontools Conclusion Limits A posteriori (history of mathematics): “just read your classics”. But the classics only describe what has been done, not necessarily what can be done. A priori (computer science): “just make a machine”. But we lose the anthropological side of the problem. Idealized computers are not computing men. (And infinite abaci do not exist.) 5/51 BaptisteMélès ProgrammingLanguagesforCalculatingTools Introduction “Manisananimalwhothinkswithhisfingers” Abacus Historyofmathematicsandcomputerscience Counting-rods Anexampleofstructuralanthropology: percussiontools Conclusion Structural anthropology and technology To answer our question, we would need to mix those two approches — anthropology and a priori science. But is it possible? Yes, it seems to be! There is at least one example. The idea would be to have something like that: 6/51 BaptisteMélès ProgrammingLanguagesforCalculatingTools
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